Mining the geodesic equation for scattering data

The geodesic equation encodes test-particle dynamics at arbitrary gravitational coupling, hence retaining all orders in the post-Minkowskian (PM) expansion. Here we explore what motion can tell us about dynamical scattering presence of perturbatively small effects such as tidal distortion and higher derivative corrections to general relativity. We derive an algebraic map between perturbed leading PM amplitude mass ratio. As examples, compute formulas for amplitudes isotropic gauge Hamiltonians certain infinite classes operators electric or magnetic Weyl any power, gravitationally interacting bodies with without charge. Finally, present a method calculating closed-form expressions limit